Equipartition theorem

thumb|right|Thermal motion of an [[alpha helix|α-helical peptide. The jittery motion is random and complex, and the energy of any particular atom can fluctuate wildly. Nevertheless, the equipartition theorem allows the average kinetic energy of each atom to be computed, as well as the average potential energies of many vibrational modes. The grey, red and blue spheres represent atoms of carbon, oxygen and nitrogen, respectively; the smaller white spheres represent atoms of hydrogen.]]

In classical statistical mechanics, the equipartition theorem relates the temperature of a system to its average energies. The equipartition theorem is also known as the law of equipartition, equipartition of energy, or simply equipartition. The original idea of equipartition was that, in thermal equilibrium, energy is shared equally among all of its various forms; for example, the average kinetic energy per degree of freedom in translational motion of a molecule should equal that in rotational motion.

The equipartition theorem makes quantitative predictions. Like the virial theorem, it gives the total average kinetic and potential energies for a system at a given temperature, from which the system's heat capacity can be computed. However, equipartition also gives the average values of individual components of the energy, such as the kinetic energy of a particular particle or the potential energy of a single spring. For example, it predicts that every atom in a monatomic ideal gas has an average kinetic energy of in thermal equilibrium, where is the Boltzmann constant and T is the (thermodynamic) temperature. More generally, equipartition can be applied to any classical system in thermal equilibrium, no matter how complicated. It can be used to derive the ideal gas law, and the Dulong–Petit law for the specific heat capacities of solids. The equipartition theorem can also be used to predict the properties of stars, even white dwarfs and neutron stars, since it holds even when relativistic effects are considered.

Although the equipartition theorem makes accurate predictions in certain conditions, it is inaccurate when quantum effects are significant, such as at low temperatures. When the thermal energy is smaller than the quantum energy spacing in a particular degree of freedom, the average energy and heat capacity of this degree of freedom are less than the values predicted by equipartition. Such a degree of freedom is said to be "frozen out" when the thermal energy is much smaller than this spacing. For example, the heat capacity of a solid decreases at low temperatures as various types of motion become frozen out, rather than remaining constant as predicted by equipartition. Such decreases in heat capacity were among the first signs to physicists of the 19th century that classical physics was incorrect and that a new, more subtle, scientific model was required. Along with other evidence, equipartition's failure to model black-body radiation—also known as the ultraviolet catastrophe—led Max Planck to suggest that energy in the oscillators in an object, which emit light, were quantized, a revolutionary hypothesis that spurred the development of quantum mechanics and quantum field theory.

Basic concept and simple examples

right|thumb|upright=1.7|Figure 2. Probability density functions of the molecular speed for four [[noble gases at a temperature of 298.15 K (25 °C). The four gases are helium (4He), neon (20Ne), argon (40Ar) and xenon (132Xe); the superscripts indicate their mass numbers. These probability density functions have dimensions of probability times inverse speed; since probability is dimensionless, they can be expressed in units of seconds per meter.]]

The name "equipartition" means "equal division," as derived from the Latin equi from the antecedent, æquus ("equal or even"), and partition from the noun, partitio ("division, portion"). The original concept of equipartition was that the total kinetic energy of a system is shared equally among all of its independent parts, on the average, once the system has reached thermal equilibrium. Equipartition also makes quantitative predictions for these energies. For example, it predicts that every atom of an inert noble gas, in thermal equilibrium at temperature , has an average translational kinetic energy of , where is the Boltzmann constant. As a consequence, since kinetic energy is equal to (mass)(velocity)2, the heavier atoms of xenon have a lower average speed than do the lighter atoms of helium at the same temperature. Figure 2 shows the Maxwell–Boltzmann distribution for the speeds of the atoms in four noble gases.

In this example, the key point is that the kinetic energy is quadratic in the velocity. The equipartition theorem shows that in thermal equilibrium, any degree of freedom (such as a component of the position or velocity of a particle) which appears only quadratically in the energy has an average energy of and therefore contributes to the system's heat capacity. This has many applications.

Translational energy and ideal gases

The (Newtonian) kinetic energy of a particle of mass , velocity is given by

<math display="block">H_{\text{kin = \tfrac 1 2 m |\mathbf{v}|^2 = \tfrac{1}{2} m\left( v_x^2 + v_y^2 + v_z^2 \right),</math>

where , and are the Cartesian components of the velocity . Here, is short for Hamiltonian, and used henceforth as a symbol for energy because the Hamiltonian formalism plays a central role in the most general form of the equipartition theorem.

Since the kinetic energy is quadratic in the components of the velocity, by equipartition these three components each contribute to the average kinetic energy in thermal equilibrium. Thus the average kinetic energy of the particle is , as in the example of noble gases above.

More generally, in a monatomic ideal gas the total energy consists purely of (translational) kinetic energy: by assumption, the particles have no internal degrees of freedom and move independently of one another. Equipartition therefore predicts that the total energy of an ideal gas of particles is .

It follows that the heat capacity of the gas is and hence, in particular, the heat capacity of a mole of such gas particles is , where NA is the Avogadro constant and R is the gas constant. Since R ≈ 2 cal/(mol·K), equipartition predicts that the molar heat capacity of an ideal gas is roughly 3 cal/(mol·K). This prediction is confirmed by experiment when compared to monatomic gases.

Rotational energy and molecular tumbling in solution

A similar example is provided by a rotating molecule with principal moments of inertia , and . According to classical mechanics, the rotational energy of such a molecule is given by

<math display="block">H_{\mathrm{rot = \tfrac{1}{2} ( I_1 \omega_1^2 + I_2 \omega_2^2 + I_3 \omega_3^2 ),</math>

where , , and are the principal components of the angular velocity. By exactly the same reasoning as in the translational case, equipartition implies that in thermal equilibrium the average rotational energy of each particle is . Similarly, the equipartition theorem allows the average (more precisely, the root mean square) angular speed of the molecules to be calculated. Rotational diffusion can also be observed by other biophysical probes such as fluorescence anisotropy, flow birefringence and dielectric spectroscopy.

Potential energy and harmonic oscillators

Equipartition applies to potential energies as well as kinetic energies: important examples include harmonic oscillators such as a spring, which has a quadratic potential energy

<math display="block">H_{\text{pot = \tfrac 1 2 a q^2,\,</math>

where the constant describes the stiffness of the spring and is the deviation from equilibrium. If such a one-dimensional system has mass , then its kinetic energy is

where and denote the velocity and momentum of the oscillator. Combining these terms yields the total energy and the Dulong–Petit law of solid heat capacities. The latter application was particularly significant in the history of equipartition.

frame|left|Figure 3. Atoms in a crystal can vibrate about their equilibrium positions in the [[crystal structure|lattice. Such vibrations account largely for the heat capacity of crystalline dielectrics; with metals, electrons also contribute to the heat capacity.]]

Specific heat capacity of solids

An important application of the equipartition theorem is to the specific heat capacity of a crystalline solid. Each atom in such a solid can oscillate in three independent directions, so the solid can be viewed as a system of independent simple harmonic oscillators, where denotes the number of atoms in the lattice. Since each harmonic oscillator has average energy , the average total energy of the solid is , and its heat capacity is .

By taking to be the Avogadro constant , and using the relation between the gas constant and the Boltzmann constant , this provides an explanation for the Dulong–Petit law of specific heat capacities of solids, which stated that the specific heat capacity (per unit mass) of a solid element is inversely proportional to its atomic weight. A modern version is that the molar heat capacity of a solid is 3R ≈ 6 cal/(mol·K).

However, this law is inaccurate at lower temperatures, due to quantum effects; it is also inconsistent with the experimentally derived third law of thermodynamics, according to which the molar heat capacity of any substance must go to zero as the temperature goes to absolute zero. Over time, these clumps settle downwards under the influence of gravity, causing more haze near the bottom of a bottle than near its top. However, in a process working in the opposite direction, the particles also diffuse back up towards the top of the bottle. Once equilibrium has been reached, the equipartition theorem may be used to determine the average position of a particular clump of buoyant mass . For an infinitely tall bottle of beer, the gravitational potential energy is given by

<math display="block">H^{\mathrm{grav = m_\text{b} g z </math>

where is the height of the protein clump in the bottle and g is the acceleration due to gravity. Since , the average potential energy of a protein clump equals . Hence, a protein clump with a buoyant mass of 10 MDa (roughly the size of a virus) would produce a haze with an average height of about 2 cm at equilibrium. The process of such sedimentation to equilibrium is described by the Mason–Weaver equation.

History

The equipartition of kinetic energy was proposed initially in 1843, and more correctly in 1845, by John James Waterston. In 1859, James Clerk Maxwell argued that the kinetic heat energy of a gas is equally divided between linear and rotational energy. In 1876, Ludwig Boltzmann expanded on this principle by showing that the average energy was divided equally among all the independent components of motion in a system. Boltzmann applied the equipartition theorem to provide a theoretical explanation of the Dulong–Petit law for the specific heat capacities of solids.

thumb|upright=1.5|Figure 4. Idealized plot of the [[specific heat capacity|molar specific heat of a diatomic gas against temperature. It agrees with the value (7/2)R predicted by equipartition at high temperatures (where R is the gas constant), but decreases to (5/2)R and then R at lower temperatures, as the vibrational and rotational modes of motion are "frozen out". The failure of the equipartition theorem led to a paradox that was only resolved by quantum mechanics. For most molecules, the transitional temperature Trot is much less than room temperature, whereas Tvib can be ten times larger or more. A typical example is carbon monoxide, CO, for which Trot ≈ 2.8 K and Tvib ≈ 3103 K. For molecules with very large or weakly bound atoms, Tvib can be close to room temperature (about 300 K); for example, Tvib ≈ 308 K for iodine gas, I2. Their law was used for many years as a technique for measuring atomic weights. However, subsequent studies by James Dewar and Heinrich Friedrich Weber showed that this Dulong–Petit law holds only at high temperatures; at lower temperatures, or for exceptionally hard solids such as diamond, the specific heat capacity was lower.

Experimental observations of the specific heat capacities of gases also raised concerns about the validity of the equipartition theorem. The theorem predicts that the molar heat capacity of simple monatomic gases should be roughly 3 cal/(mol·K), whereas that of diatomic gases should be roughly 7 cal/(mol·K). Experiments confirmed the former prediction, but found that molar heat capacities of diatomic gases were typically about 5 cal/(mol·K), and fell to about 3 cal/(mol·K) at very low temperatures. Maxwell noted in 1875 that the disagreement between experiment and the equipartition theorem was much worse than even these numbers suggest; since atoms have internal parts, heat energy should go into the motion of these internal parts, making the predicted specific heats of monatomic and diatomic gases much higher than 3 cal/(mol·K) and 7 cal/(mol·K), respectively.

A third discrepancy concerned the specific heat of metals. According to the classical Drude model, metallic electrons act as a nearly ideal gas, and so they should contribute to the heat capacity by the equipartition theorem, where Ne is the number of electrons. Experimentally, however, electrons contribute little to the heat capacity: the molar heat capacities of many conductors and insulators are nearly the same. Lord Kelvin suggested that the derivation of the equipartition theorem must be incorrect, since it disagreed with experiment, but was unable to show how. In 1900 Lord Rayleigh instead put forward a more radical view that the equipartition theorem and the experimental assumption of thermal equilibrium were both correct; to reconcile them, he noted the need for a new principle that would provide an "escape from the destructive simplicity" of the equipartition theorem. Albert Einstein provided that escape, by showing in 1906 that these anomalies in the specific heat were due to quantum effects, specifically the quantization of energy in the elastic modes of the solid. Einstein used the failure of equipartition to argue for the need of a new quantum theory of matter. supported Einstein's theory, and led to the widespread acceptance of quantum theory among physicists.

General formulation of the equipartition theorem

The most general form of the equipartition theorem states that under suitable assumptions (discussed below), for a physical system with Hamiltonian energy function and degrees of freedom , the following equipartition formula holds in thermal equilibrium for all indices and :

<math display="block">\left\langle x_{m} \frac{\partial H}{\partial x_{n \right\rangle = \delta_{mn} k_\text{B} T.</math>

Here is the Kronecker delta, which is equal to one if and is zero otherwise. The averaging brackets <math>\left\langle \ldots \right\rangle</math> is assumed to be an ensemble average over phase space or, under an assumption of ergodicity, a time average of a single system.

The general equipartition theorem holds in both the microcanonical ensemble, when the total energy of the system is constant, and also in the canonical ensemble, when the system is coupled to a heat bath with which it can exchange energy. Derivations of the general formula are given later in the article.

The general formula is equivalent to the following two:

<math>\left\langle x_n \frac{\partial H}{\partial x_n} \right\rangle = k_\text{B} T \quad \text{for all } n</math>
<math>\left\langle x_m \frac{\partial H}{\partial x_n} \right\rangle = 0 \quad \text{for all } m \neq n.</math>

If a degree of freedom xn appears only as a quadratic term anxn2 in the Hamiltonian H, then the first of these formulae implies that

<math display="block">k_\text{B} T = \left\langle x_n \frac{\partial H}{\partial x_n}\right\rangle = 2\left\langle a_n x_n^2 \right\rangle,</math>

which is twice the contribution that this degree of freedom makes to the average energy <math>\langle H\rangle</math>. Thus the equipartition theorem for systems with quadratic energies follows easily from the general formula. A similar argument, with 2 replaced by s, applies to energies of the form anxns.

The degrees of freedom xn are coordinates on the phase space of the system and are therefore commonly subdivided into generalized position coordinates qk and generalized momentum coordinates pk, where pk is the conjugate momentum to qk. In this situation, formula 1 means that for all k,

\left\langle p_{k} \frac{\partial H}{\partial p_{k \right\rangle = \left\langle q_k \frac{\partial H}{\partial q_k} \right\rangle = k_\text{B} T.

</math>

Using the equations of Hamiltonian mechanics, these formulae may also be written

<math display="block">\left\langle p_k \frac{dq_k}{dt} \right\rangle = -\left\langle q_k \frac{dp_k}{dt} \right\rangle = k_\text{B} T.</math>

Similarly, one can show using formula 2 that

\left\langle p_j \frac{\partial H}{\partial p_k} \right\rangle =

\left\langle q_j \frac{\partial H}{\partial q_k} \right\rangle = 0 \quad \text{ for all } \, j \neq k.

</math>

and

\left\langle p_j \frac{\partial q_k}{\partial t} \right\rangle =

-\left\langle q_j \frac{\partial p_k}{\partial t} \right\rangle = 0 \quad \text{ for all } \, j \neq k.

</math>

Relation to the virial theorem

The general equipartition theorem is an extension of the virial theorem (proposed in 1870), which states that

\left\langle \sum_k q_k \frac{\partial H}{\partial q_{k \right\rangle =

\left\langle \sum_k p_k \frac{\partial H}{\partial p_{k \right\rangle =

\left\langle \sum_k p_k \frac{dq_k}{dt} \right\rangle = -\left\langle \sum_k q_k \frac{dp_k}{dt} \right\rangle,

</math>

where t denotes time.

Diatomic gases

A diatomic gas can be modelled as two masses, and , joined by a spring of stiffness , which is called the rigid rotor-harmonic oscillator approximation. The classical energy of this system is

<math display="block">H =

\frac{\left| \mathbf{p}_1 \right|^2}{2m_1} +

\frac{\left| \mathbf{p}_2 \right|^2}{2m_2} +

\frac{1}{2} a q^2,</math>

where and are the momenta of the two atoms, and is the deviation of the inter-atomic separation from its equilibrium value. Every degree of freedom in the energy is quadratic and, thus, should contribute to the total average energy, and to the heat capacity. Therefore, the heat capacity of a gas of N diatomic molecules is predicted to be : the momenta and contribute three degrees of freedom each, and the extension contributes the seventh. It follows that the heat capacity of a mole of diatomic molecules with no other degrees of freedom should be and, thus, the predicted molar heat capacity should be roughly 7 cal/(mol·K). However, the experimental values for molar heat capacities of diatomic gases are typically about 5 cal/(mol·K) It follows that the mean potential energy associated to the interaction of the given particle with the rest of the gas is

<math display="block">\langle h_{\mathrm{pot \rangle = \int_0^\infty 4\pi r^2 \rho U(r) g(r)\, dr.</math>

The total mean potential energy of the gas is therefore <math> \langle H_\text{pot} \rangle = \tfrac12 N \langle h_{\mathrm{pot \rangle </math>, where is the number of particles in the gas, and the factor is needed because summation over all the particles counts each interaction twice.

Adding kinetic and potential energies, then applying equipartition, yields the energy equation

<math display="block">H = \langle H_{\mathrm{kin \rangle + \langle H_{\mathrm{pot \rangle =

\frac{3}{2} Nk_\text{B}T + 2\pi N \rho \int_0^\infty r^2 U(r) g(r) \, dr.</math>

A similar argument, Simple examples are provided by potential energy functions of the form

<math display="block">H_{\mathrm{pot = C q^{s},\,</math>

where and are arbitrary real constants. In these cases, the law of equipartition predicts that

k_\text{B} T = \left\langle q \frac{\partial H_{\mathrm{pot}{\partial q} \right\rangle =

\langle q \cdot s C q^{s-1} \rangle = \langle s C q^{s} \rangle = s \langle H_{\mathrm{pot \rangle.

</math>

Thus, the average potential energy equals , not as for the quadratic harmonic oscillator (where ).

More generally, a typical energy function of a one-dimensional system has a Taylor expansion in the extension :

<math display="block">H_{\mathrm{pot = \sum_{n=2}^\infty C_n q^n</math>

for non-negative integers . There is no term, because at the equilibrium point, there is no net force and so the first derivative of the energy is zero. The term need not be included, since the energy at the equilibrium position may be set to zero by convention. In this case, the law of equipartition predicts that As examples, the virial theorem may be used to estimate stellar temperatures or the Chandrasekhar limit on the mass of white dwarf stars.

The average temperature of a star can be estimated from the equipartition theorem. Since most stars are spherically symmetric, the total gravitational potential energy can be estimated by integration

<math display="block">H_{\mathrm{grav = -\int_0^R \frac{4\pi r^2 G}{r} M(r)\, \rho(r)\, dr,</math>

where is the mass within a radius and is the stellar density at radius ; represents the gravitational constant and the total radius of the star. Assuming a constant density throughout the star, this integration yields the formula

<math display="block">H_{\mathrm{grav = - \frac{3G M^{2{5R},</math>

where is the star's total mass. Hence, the average potential energy of a single particle is

<math display="block">\langle H_{\mathrm{grav \rangle = \frac{H_{\mathrm{grav}{N} = - \frac{3G M^{2{5RN},</math>

where is the number of particles in the star. Since most stars are composed mainly of ionized hydrogen, equals roughly , where is the mass of one proton. Application of the equipartition theorem gives an estimate of the star's temperature

<math display="block">\left\langle r \frac{\partial H_{\mathrm{grav}{\partial r} \right\rangle = \langle -H_{\mathrm{grav \rangle = k_\text{B} T = \frac{3G M^2}{5RN}.</math>

Substitution of the mass and radius of the Sun yields an estimated solar temperature of T = 14 million kelvins, very close to its core temperature of 15 million kelvins. However, the Sun is much more complex than assumed by this model—both its temperature and density vary strongly with radius—and such excellent agreement (≈7% relative error) is partly fortuitous.

Star formation

The same formulae may be applied to determining the conditions for star formation in giant molecular clouds. A local fluctuation in the density of such a cloud can lead to a runaway condition in which the cloud collapses inwards under its own gravity. Such a collapse occurs when the equipartition theorem—or, equivalently, the virial theorem—is no longer valid, i.e., when the gravitational potential energy exceeds twice the kinetic energy

Assuming a constant density for the cloud

yields a minimum mass for stellar contraction, the Jeans mass

<math display="block">M_\text{J}^{2} = \left( \frac{5k_\text{B}T}{G m_{p \right)^{3} \left( \frac{3}{4\pi \rho} \right).</math>

Substituting the values typically observed in such clouds (, ) gives an estimated minimum mass of 17 solar masses, which is consistent with observed star formation. This effect is also known as the Jeans instability, after the British physicist James Hopwood Jeans who published it in 1902.

Derivations

Kinetic energies and the Maxwell–Boltzmann distribution

The original formulation of the equipartition theorem states that, in any physical system in thermal equilibrium, every particle has exactly the same average translational kinetic energy, . However, this is true only for ideal gas, and the same result can be derived from the Maxwell–Boltzmann distribution. First, we choose to consider only the Maxwell–Boltzmann distribution of velocity of the z-component

<math display="block">f (v_z) = \sqrt{\dfrac{m}{2\pi k_\text{B}T\;e^{\frac{-m{v_z}^2}{2k_\text{B}T</math>

with this equation, we can calculate the mean square velocity of the -component

<math display="block">\langle {v_z}^2 \rangle = \int_{-\infty}^{\infty} f (v_z){v_z}^2 dv_z = \dfrac{k_\text{B}T}{m}</math>

Since different components of velocity are independent of each other and share the same distribution function, the average translational kinetic energy is given by

<math display="block">\langle E_k \rangle = \dfrac 3 2 m \langle {v_z}^2 \rangle = \dfrac 3 2 k_\text{B}T</math>

Notice, the Maxwell–Boltzmann distribution should not be confused with the Boltzmann distribution, which the former can be derived from the latter by assuming the energy of a particle is equal to its translational kinetic energy.

As stated by the equipartition theorem. The same result can also be obtained by averaging the particle energy using the probability of finding the particle in certain quantum energy state. Integration over the variable yields a factor

<math display="block">Z_{x} = \int_{-\infty}^{\infty} dx \ e^{-\beta A x^{2 = \sqrt{\frac{\pi}{\beta A,</math>

in the formula for . The mean energy associated with this factor is given by

<math display="block">\langle H_{x} \rangle = - \frac{\partial \log Z_{x{\partial \beta} = \frac{1}{2\beta} = \frac{1}{2} k_\text{B} T</math>

as stated by the equipartition theorem.

General proofs

General derivations of the equipartition theorem can be found in many statistical mechanics textbooks, both for the microcanonical ensemble The requirements for isolated systems to ensure ergodicity—and, thus equipartition—have been studied, and provided motivation for the modern chaos theory of dynamical systems. A chaotic Hamiltonian system need not be ergodic, although that is usually a good assumption. If the system is isolated from the rest of the world, the energy in each normal mode is constant; energy is not transferred from one mode to another. Hence, equipartition does not hold for such a system; the amount of energy in each normal mode is fixed at its initial value. If sufficiently strong nonlinear terms are present in the energy function, energy may be transferred between the normal modes, leading to ergodicity and rendering the law of equipartition valid. However, the Kolmogorov–Arnold–Moser theorem states that energy will not be exchanged unless the nonlinear perturbations are strong enough; if they are too small, the energy will remain trapped in at least some of the modes.

Another simple example is an ideal gas of a finite number of colliding particles in a round vessel. Due to the vessel's symmetry, the angular momentum of such a gas is conserved. Therefore, not all states with the same energy are populated. This results in the mean particle energy being dependent on the mass of this particle, and also on the masses of all the other particles.

Another way ergodicity can be broken is by the existence of nonlinear soliton symmetries. In 1953, Fermi, Pasta, Ulam and Tsingou conducted computer simulations of a vibrating string that included a non-linear term (quadratic in one test, cubic in another, and a piecewise linear approximation to a cubic in a third). They found that the behavior of the system was quite different from what intuition based on equipartition would have led them to expect. Instead of the energies in the modes becoming equally shared, the system exhibited a very complicated quasi-periodic behavior. This puzzling result was eventually explained by Kruskal and Zabusky in 1965 in a paper which, by connecting the simulated system to the Korteweg–de Vries equation led to the development of soliton mathematics.

Failure due to quantum effects

The law of equipartition breaks down when the thermal energy is significantly smaller than the spacing between energy levels. Equipartition no longer holds because it is a poor approximation to assume that the energy levels form a smooth continuum, which is required in the derivations of the equipartition theorem above. The paradox arises because there are an infinite number of independent modes of the electromagnetic field in a closed container, each of which may be treated as a harmonic oscillator. If each electromagnetic mode were to have an average energy , there would be an infinite amount of energy in the container. However, by the reasoning above, the average energy in the higher-frequency modes goes to zero as ν goes to infinity; moreover, Planck's law of black-body radiation, which describes the experimental distribution of energy in the modes, follows from the same reasoning.